Gary and Elaine are competing as a team in an orienteering event. They follow instructions telling them to go 50 m [N], 75 m [NE] and 100 m [30o E of S].
a) Calculate the total distance travelled
b) Draw a vector diagram to determine the resultant displacement.
To calculate the total distance traveled by Gary and Elaine, we need to add up the distances they traveled in each direction.
a) To calculate the total distance traveled, we can use the Pythagorean theorem. The distances they traveled in each direction can be considered as the sides of a right triangle. Let's calculate:
- 50 m [N]: This is the vertical component of their displacement, so we can write it as 50 m * sin(90°) = 50 m.
- 75 m [NE]: This is a diagonal displacement, so we can use the horizontal and vertical components of this displacement. The angle between the diagonal and vertical components is 45°, so the vertical component is 75 m * sin(45°) = 53.03 m.
- 100 m [30° E of S]: This is another diagonal displacement, but at a different angle. We need to calculate the vertical component. The angle between the diagonal and vertical components is 60° (since it is 30° east of the south, which is a 30° angle from the vertical), so the vertical component is 100 m * sin(60°) ≈ 86.60 m.
Now we add up the vertical components:
Total Vertical Distance = 50 m + 53.03 m + 86.60 m ≈ 189.63 m
b) To determine the resultant displacement, we need to add up the vectors graphically. We can draw a vector diagram to visualize this.
- First, draw the vector for the displacement of 50 m [N]. It should be a straight line pointing upward.
- Next, draw the vector for the displacement of 75 m [NE]. It should be a diagonal line pointing northeast from the tip of the first vector.
- Finally, draw the vector for the displacement of 100 m [30° E of S]. It should be a diagonal line pointing 30° east of south from the tip of the second vector.
The resultant displacement is the vector that starts at the origin and ends at the tip of the last vector. Measure the length and direction of this resultant vector to determine the displacement.
I hope this helps you calculate the total distance traveled and draw the vector diagram accurately!
a. d = 50 + 75 + 100 =
b. D = 50i + 75[45o] + 100[300o] = 50i + 53+53i + 50-86.6i = 103 - 33.6i = 108.3m[-18.1o] = 18.1o S. of E.